Talk:Molecular Hamiltonian

Definitions
Hmm...Merge with Electronic Hamiltonian? --HappyCamper 20:43, 23 September 2006 (UTC)


 * They describe the same things, and a merge with the Born-Oppenheimer_approximation and expandning the merged article to include parts about the vibronic and rotational Hamiltonian would make the articles much more usefull--Martin Hedegaard 22.19, 24 September 2006 (UTC)


 * I agree. Actually, these articles on Wikipedia are a mess right now, and really need some major cleanup. Maybe in a few days we can merge? --HappyCamper 18:29, 24 September 2006 (UTC)


 * I be happy to help to expand the resulting article and I have quite good references on the subject, but I am still a little shaky in the workings of wikipedia so a little help in this merge project would be appreciated. I my opinion the best would be to expand Electronic Hamiltonian to include the information given in Molecular Hamiltonian and the Born-Oppenheimer_approximation and expand the subject first Martin Hedegaard 14:56, 25 September 2006 (UTC)


 * Sure, there is some information at Merging and moving pages which you might find useful. I agree with your approach too. Is everything on this page in Electronic Hamiltonian now? If so, we can blank this page and make it into a redirect. I also see you have a sandbox set up. I can merge that with the existing electronic Hamiltonian article once its done if you like. --HappyCamper 03:53, 26 September 2006 (UTC)


 * I dont think that the article on Electronic Hamiltonian is covering the concept of Molecular Hamiltonian very well right now, but I am working on it in my sandbox right now. When it is finished it would probably replace both Molecular Hamiltonian and Born-Oppenheimer_approximation, and add information about the rotational and vibrational Hamiltonian it just need a bit more work and a new intro part -- Martin Hedegaard 09:01, 26 September 2006 (UTC)


 * Made a big rework on the article in my sandbox that should cover most of the Molecular Hamiltonian, and Electronic Hamiltonian and Molecular Hamiltonian should probably be merged under the name of Molecular Hamiltonian. The article Born-Oppenheimer_approximation should probably go there too, but at proper explanation of the concept involves description of the splitting of the wavefunction, and the crude born oppenheimer approximation Martin Hedegaard 11:34, 26 September 2006 (UTC)


 * Yes, I noticed that :-) In that case, we can simply move the sandbox contents over to the main article. Don't cut and paste it though, because it's important that we keep the edit histories which accompany the articles. --HappyCamper 16:23, 26 September 2006 (UTC)


 * I would like a breif review of the sandbox before moving it, I am not a native english speaker, and its a long time since I wrote anything english. If its okay just go ahead and move the content. The Electronic Hamiltonian should just be a redirect to this page, the new article covers exactly the same things as the old article-- Martin Hedegaard 16:45, 26 September 2006 (UTC)


 * Okay, well here is the result after some swapping around. If it turns out this is problematic, we can fix it. --HappyCamper 20:54, 26 September 2006 (UTC)

Born-Oppenheimer
I would like to separate off the BO lemma again. I have seen the criticisms, especially by the anonymous from the University of Stockholm (130.237.179.166). I understand what (s)he is saying and I can write an article that will not offend him/her too much. Anybody has any objections? --P.wormer 15:31, 8 December 2006 (UTC)


 * Nope, feel free to edit as you see fit. --HappyCamper 04:33, 10 December 2006 (UTC)

Unusual notation
The notation used in this article:
 * $$ \frac{\partial}{\partial R_i}$$

is unusual for differentiation with respect to a vector. More common is either $$\nabla_i$$ or introduction of components
 * $$ \frac{\partial}{\partial R_{i\alpha}} $$

with $$ \alpha = x,y,z $$. Is this notation on purpose, or just by mistake?--P.wormer 10:56, 26 December 2006 (UTC)
 * I have seen that kind of notation before, but I think that we should go with the $$\nabla$$ notation because its more consistent with the rest of the article, and maybe give a link to a page like Nabla_in_cylindrical_and_spherical_coordinates. The part you are reffering to is one of the last parts of the orginal article thats why the notation is inconsistent, anyway its because of any special reason. Martin Hedegaard 16:08, 26 December 2006 (UTC)

Overhaul
I was not very satisfied with this lemma, so I give it a new try. I started today, but will continue.

Question: why is the article by Handy et al. included in the reference list? In my view it is just one of the many research papers written about different terms in the Hamiltonian. This one is about a specific computational method for the BO diagonal correction and continues similar work by others, e.g. by David Yarkoni. Several of the review papers by Brian Sutcliffe would be more appropriate for the reference list, I would think.--P.wormer 14:02, 1 January 2007 (UTC)


 * Its there because it was a part of the old aticles, when I rewrote it the first time and I just let i be, if you have better references or some good review papers then go ahead and fill it in. I actually thought the same thing about the paper from Handy et al., but it was not wrong as far as i could see, and referes to the last part of the article, that I dident write.


 * If you have any specific references in mind, please post them so we can find the best possible references for this lemma. Martin Hedegaard 14:36, 1 January 2007 (UTC)

Too many bold letters?
I wonder if it would be better to replace our bold letters with vector notation instead? The presentation comes across as a bit heavy, I think. Thoughts? --HappyCamper 03:00, 5 January 2007 (UTC)
 * You mean with arrows on top? I heard that an SI committee advised sans serif letters for matrices and vectors, would that be an idea? --P.wormer 10:10, 5 January 2007 (UTC)
 * Wow, really? That's news to me. Okay, we'll not use the arrows and keep what we have already. --HappyCamper 11:00, 9 January 2007 (UTC)

Finished overhaul
Basically I finished the overhaul of the article. I thought I knew this stuff, but writing it I'm amazed how many holes there are in this theory. So I had to skim along WP:NOR. See also Talk:GF_method. I commented out the last part of the original text, so if somebody feels that (some of) it must be restituted, please uncomment it. --P.wormer 16:06, 10 January 2007 (UTC)


 * What do you mean by skimming along WP:NOR? --HappyCamper 12:09, 13 January 2007 (UTC)


 * Well, some sentences reasonings are not copied straight from a book, but thought up by me and I hope I didn't make mistakes. I'm still thinking and reading. --P.wormer 15:11, 13 January 2007 (UTC)


 * Maybe we need something like the introduction in H. Köppel, W. Domcke, and L. Cederbaum, Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation, Adv. Chem. Phys., vol. 57, 1984. I don't have this with me right now, but I vaguely remember there is a nice concise way of summarizing things there. --HappyCamper 15:21, 13 January 2007 (UTC)

As far as I remember those guys write about conical intersections and such things, that is break-down  of Born-Oppenheimer. So, their work is relevant for the diabatic and Born-Oppenheimer articles. If you see any room for improvements in those articles, please edit them in. For the present article Molecular Hamiltonian my problems have to do with (i) nuclear mass polarization: why do many authors (Wilson&Decius&Cross and Papousek&Aliev and Louck) not discuss them? They should get them by transforming to the nuclear center of mass. (ii) Internal versus external coordinates. Wilson&Decius&Cross define linearized valence coordinates without mass weighting. These are internal coordinates. In two other chapters they define internal coordinates with mass weighting (via the Eckart conditions). The Watson nuclear motion Hamiltonian is defined with Eckart conditions. How do Wilson's linearized coordinates fit into this? If anybody knows the answers to this, or knows literature that contains the answers, or sees that I made error(s) here (s)he should not hesitate to edit. --P.wormer 17:54, 13 January 2007 (UTC)

Redirects
Just thought I'd point out that Born-Oppenheimer Approximation (note the capital A) redirects here, even though it has its own article. My guess is that it got lost in the recent overhaul. I'd fix it myself, but I really don't know where to start. Gershwinrb 08:34, 19 January 2007 (UTC)
 * I have fixed it. Born-Oppenheimer Approximation redirect now to Born-Oppenheimer approximation. --Bduke 11:29, 19 January 2007 (UTC)

Potential energy surface(s)
The collection of electronic energies for varying nuclear coordinates R forms one potential energy surface that enters one nuclear Schroedinger eq.--P.wormer 15:00, 22 January 2007 (UTC)


 * Of course not! There are many eigenvalues! Vb 09:44, 23 January 2007 (UTC)
 * Vb: I spent lots of time and thoughts on this article, so don't be rash and pigheaded. Please read the Born-Oppenheimer article. Within the BO approximation one goes from the Coulomb (electronic) Hamiltonian to the nuclear motion Hamiltonian with the use of ONE potential energy surface. Of course, the Coulomb Hamiltonian has more (often infinitely many) excited states and energies, but the whole idea is that one considers only ONE state with ONE energy (almost always this is the ground state and its energy) for MANY nuclear constellations. Please, if you don't understand this point, leave this article alone. Thank you. --P.wormer 13:19, 23 January 2007 (UTC)


 * I am sorry but I spent enough time computing sets of potential surfaces to know that the BO idea applies to many coupled potential surfaces. That nowadays most people are most concerned with the groundstate is clear to me.  However this article must be clear on this point! User:Vb 09:58, 1 February 2007 (UTC)

Redirected from Adiabatic Approximation
Searching for "Adiabatic Approximation" redirects to this article. This article does NOT describe the adiabatic approximation, however (it merely describes the entity that results when this approximation is made for certain physical systems).

Should this redirect be eliminated? —Preceding unsigned comment added by Kaiserkarl13 (talk • contribs) 14:25, 30 April 2008 (UTC)


 * I agree that "adiabatic approximation" should not link here. Instead, it should redirect to Adiabatic theorem.  I have made the change.  Cgd8d (talk) 20:54, 8 May 2008 (UTC)

Rewrite
The bricks and mortar analogy is lame...can we rewrite this? Punctilius (talk) 05:14, 9 September 2008 (UTC)

Rotational Hamiltonian
Pure rotational spectra are very hard to achieve experimentally, but they can be described by further separation of the vibrational and electronic motions. This requires two things:


 * 1) Assume that the nuclei only make small oscillations from equilibrium configuration so the vibrational potential can be considered harmonic;
 * 2) Approximate the inertia tensor with the inertia tensor $$I_{n,eq} \,\;$$ calculated at the equilibrium configuration.

This is also called the "Harmonic vibrational and rigid-rotor model."

Vibronic Hamiltonian
This is the most prevalent form of the molecular Hamiltonian because the vibrations are essentially independent of the surroundings. Hence, vibrational transitions are easily observed. Since the rotational transitions are almost never observed, a good approximation to the molecular Hamiltonian would be obtained by keeping only the part of HM that describes the electronic and vibrational parts. This is called the vibronic Hamiltonian, a portmanteau of "vibrational" and "electronic". The vibronic Hamiltonian is given by


 * $$\hat{H}_{M,vibronic}=\hat{T}_{e,vibronic}+\hat{T}_{n,vibronic}+V(x_e,X_n)$$

with


 * $$\hat{T}_{e,vibronic}=\frac{-\hbar^2}{2m_e}\sum_{x_e} \hat{\nabla}^2_{x_e}\quad \mathrm{and} \quad \hat{T}_{n,vibronic}=\frac{-\hbar^2}{2}\sum_{X_n} \frac{\hat{\nabla}^2_{X_n}}{M_{X_n}}$$

with the $$(x_e,X_n)$$ being internal electronic and nuclear vibration coordinates. The use of the internal coordinates is used since the coulomb interaction only depends on the relative distance between the charged particles. Since the rotational and translational motions are now separated there will be either $$3N-5$$ or $$3N-6$$ vibrations if $$N$$ is the number of nuclei, and whether the molecule is linear or nonlinear.

Solving the molecular Schrödinger equation
The molecular Schrödinger equation is given by


 * $$\hat{H}_M \psi_a(x_e,X_n)=E_a \psi_a(x_e,X_n)$$

where $$E_a$$ refers to the energy of the state $$\psi_a(x_e,X_n)$$. To solve the Schrödinger equation it is needed to decouple the motion of the nuclei and electrons. This is done by approximating the molecular wavefunction $$\psi_a(x_e,X_n)$$ to a product of the electronic wavefunction and the nuclear vibration wavefunction. This is given by


 * $$\psi_a(x_e,X_n)=\phi_e(x_e,X_n)\cdot \chi_{e,\nu} (X_n)$$

where $$e,\nu$$ is the electronic and nuclear vibration quantum number. This formulation is termed an adiabatic wavefunction.

There are two main cases used in molecular physics, a dynamic and a static type. The dynamic type the electronic wavefunctions are assumed to follow the vibrations of the nuclei. The static case uses a static reference configuration to calculate the electronic wavefunctions, this is also called the crude adiabatic approximation.

In the dynamic approximation the electronic wavefunction is defined as the solution to the electronic Schrödinger equation


 * $$\hat{H}_e \phi_e(x_e,X_n)=E_e(X_n)\phi(x_e,X_n)$$

where


 * $$\hat{H}_e=\hat{H}_M-\hat{T}_n$$

with the electronic wavefunctions found the nuclear vibrational coordinates $$X_n=\{X_1,X_2,...X_{3N-6}\}$$ or $$X_n=\{X_1,X_2,...X_{3N-5}\}$$ can be treated as parameters and the solution of the electronic Schrödinger equation then define the dependence of the electronic wavefunction and eigenvalues on the set of nuclear vibration coordinates $$X_n$$. The electronic wavefunctions defines a complete orthonomal set of functions for each $$X_n$$ so the molecular wavefunction can be expanded in the basis.


 * $$\psi_a(x_e,X_n)=\sum_{e,\nu} \phi_e(x_e,X_n)\cdot \chi_{e,\nu} (X_n)$$

using this result in the most used vibronic case, and inserting in the electronic Schrödinger equation and neglecting electronic coupling gives a new eigenvalue equation given by


 * $$(\hat{T}_{n,vibronic}+E_{e'}(X_n))\chi_{e',\nu'}(X_n)=E_{e',\nu'}\chi_{e',\nu'}(X_n)$$

where the expansion coefficients $$\chi_{e,\nu}(X_n)$$ describes the vibrational eigenfunctions and the $$E_{e}(X_n)$$ describe the vibrational potential energy. The eigenvalue, $$E_{e}(X_n)$$ is often approximated by an harmonic function for simplification.

Limitations
When the assumptions required for the adiabatic Born-Oppenheimer approximation do not hold, the approximation is said to "break down". Other approaches are needed to properly describe the system which is beyond the Born-Oppenheimer approximation.

The explicit consideration of the coupling of electronic and nuclear (vibrational) movement is known as electron-phonon coupling in extended systems such as solid state systems. In non-extended systems such as complex isolated molecules, it is known as vibronic coupling which is important in the case of avoided crossings or conical intersections.

The so-called 'diagonal Born-Oppenheimer correction' (DBOC) can be obtained as


 * $$\langle \phi_e(x_e,X_n)|T_n|\phi_e(x_e,X_n)\rangle$$

where $$T_n$$ is the nuclear kinetic energy operator and the electronic wavefunction $$\phi_e$$ is parametrically (not explicitly) dependent on the nuclear coordinates.



I moved the commented out sections from the front page to the box above. --Kkmurray (talk) 14:32, 19 December 2010 (UTC)

Born-Oppenheimer
I would like to separate off the BO lemma again. I have seen the criticisms, especially by the anonymous from the University of Stockholm (130.237.179.166). I understand what (s)he is saying and I can write an article that will not offend him/her too much. Anybody has any objections? --P.wormer 15:31, 8 December 2006 (UTC)


 * Nope, feel free to edit as you see fit. --HappyCamper 04:33, 10 December 2006 (UTC)

Unusual notation
The notation used in this article:
 * $$ \frac{\partial}{\partial R_i}$$

is unusual for differentiation with respect to a vector. More common is either $$\nabla_i$$ or introduction of components
 * $$ \frac{\partial}{\partial R_{i\alpha}} $$

with $$ \alpha = x,y,z $$. Is this notation on purpose, or just by mistake?--P.wormer 10:56, 26 December 2006 (UTC)
 * I have seen that kind of notation before, but I think that we should go with the $$\nabla$$ notation because its more consistent with the rest of the article, and maybe give a link to a page like Nabla_in_cylindrical_and_spherical_coordinates. The part you are reffering to is one of the last parts of the orginal article thats why the notation is inconsistent, anyway its because of any special reason. Martin Hedegaard 16:08, 26 December 2006 (UTC)

Mass Polarization
The section describing the separation of the COM motion from the internal motion contained several errors. In particular, the process is not "more cumbersome" quantum mechanically (QM) than it is classically. The mass polarization term that results from separating the 3 COM coordinates from the internal motion appears in both the classical and QM Hamiltonian. Furthermore, it is completely unnecessary to introduce a generalized reduced mass tensor, when the process is done by eliminating the Nth particle from the internal Hamiltonian. There were also several factors of two missing from the Hamiltonian. I'm editing the page to try to correct these errors. 99.11.197.75 (talk) 20:12, 19 February 2012 (UTC)


 * Dear 99.11.197.75, in the expression for H&prime; (first equation in the section The Schrödinger equation of the Coulomb Hamiltonian) you removed the ti from the &nabla; and replaced it by i. Further you replaced &mu;i by mi. This suggests that you believe that the sums over i and j are still over nuclei and electrons separately. However, the new coordinates ti are  linear combinations of electronic and nuclear coordinates (as is clearly stated in the text in a sentence that you, oddly enough, did not fiddle with). So, in your opinion, what mass mi is associated with given ti&thinsp;?
 * A friendly advice: get hold of Ref. 2 of the article and study that paper carefully. --P.wormer (talk) 08:50, 20 February 2012 (UTC)